Day 24

Math 216: Statistical Thinking

Bastola

Systematic Hypothesis Testing: The Complete Framework

Key Question: How can we systematically evaluate claims about population parameters using a rigorous statistical framework? The 8-step hypothesis testing process provides a comprehensive methodology for making objective decisions with quantified uncertainty!

Steps for Hypothesis Testing Part I:

  1. State the Hypotheses
    • Null Hypothesis (\(H_0\)): No effect or no difference, posits that \(\mu = \mu_0\).
    • Alternative Hypothesis (\(H_A\)): Asserts a specific claim about the population mean \(\mu\) that is different from \(\mu_0\) (e.g., \(\mu > \mu_0\), \(\mu < \mu_0\), or \(\mu \neq \mu_0\)).
  2. Set the Significance Level (\(\alpha\))
    • The probability of making a Type I error (rejecting a true null hypothesis). Common values are 0.05, 0.01, or 0.10.
  3. Assumptions
    • The sample is a simple random sample.
    • The sample size is large (usually \(n \geq 30\)), which justifies the use of the Central Limit Theorem.
  4. Collect Data
    • Obtain a random sample and compute the sample mean (\(\bar{X}\)) and standard deviation (\(S\)).

Steps for Hypothesis Testing Part II:

  1. Calculate the Test Statistic (\(z\))

    • If the population standard deviation (\(\sigma\)) is known: \[ z = \frac{\bar{X} - \mu_0}{\sigma/\sqrt{n}} \]
    • If \(\sigma\) is unknown and \(n \geq 30\), use \(s\) (sample standard deviation): \[ z = \frac{\bar{X} - \mu_0}{s/\sqrt{n}} \]

  2. Decision Rule

    • For a one-tailed test: Reject \(H_0\) if \(z\) falls in the rejection region (either \(z < -z_{\alpha}\) for left-tailed or \(z > z_{\alpha}\) for right-tailed).
    • For a two-tailed test: Reject \(H_0\) if \(|z| > z_{\alpha/2}\).

  3. Compute the P-Value

    • The probability of observing a test statistic as extreme as, or more extreme than, the statistic calculated from the sample data, assuming that \(H_0\) is true.

  4. Conclusion

    • Reject \(H_0\) if the p-value is less than \(\alpha\); otherwise, do not reject \(H_0\). Report whether the evidence supports \(H_a\) based on the statistical significance.

One-Sided (Greater Than) Example

  • Scenario: A study group claims that their session increases the average test score above 75.
  • Sample Data: \(\bar{x} = 78\), \(s = 5\), \(n = 30\)
  • Test: \(H_0: \mu \leq 75\) vs. \(H_a: \mu > 75\)
  • Calculation: \(z = \frac{78 - 75}{5/\sqrt{30}} \approx 3.29\)
  • Conclusion: Reject \(H_0\) if \(z > 1.645\) (at \(\alpha = 0.05\)).

One-Sided (Less Than) Example

  • Scenario: A nutritionist claims that a new diet reduces the average calorie intake below 2200 calories/day.
  • Sample Data: \(\bar{x} = 2150\), \(s = 120\), \(n = 35\)
  • Test: \(H_0: \mu \geq 2200\) vs. \(H_a: \mu < 2200\)
  • Calculation: \(z = \frac{2150 - 2200}{120/\sqrt{35}} \approx -2.47\)
  • Conclusion: Reject \(H_0\) if \(z < -1.645\) (at \(\alpha = 0.05\)).

Two-Sided Example: Billing System Analysis

  • Scenario: An analyst claims that a new billing system changes the average monthly bill from $100.
  • Sample Data: \(\bar{x} = 105\), \(s = 10\), \(n = 40\)
  • Test: \(H_0: \mu = 100\) vs. \(H_a: \mu \neq 100\)
  • Calculation: \(z = \frac{105 - 100}{10/\sqrt{40}} \approx 3.16\)
  • Conclusion: Reject \(H_0\) if \(|z| > 1.96\) (at \(\alpha = 0.05\)).

Comprehensive Case Study: Medical Treatment Evaluation

Medical Trial Analysis:

  • 8-Step Process Application:
    • Hypotheses: \(H_0: \mu \leq 75\) vs. \(H_a: \mu > 75\)
    • Significance: \(\alpha = 0.05\)
    • Assumptions: Random sample, \(n = 50\) (CLT applies)
    • Test Statistic: \(z = \frac{82 - 75}{8/\sqrt{50}} = 6.19\)
    • P-value: \(p < 0.0001\)
    • Conclusion: Strong evidence that treatment improves outcomes